The cell edge of a fcc crystal is 100 pm and its density is `10.0 g cm^(-3)`. The number of atoms in 100 g of this crystal is

Options

A. `1 xx 10^(25)`
B. `2 xx 10^(25)`
C. `3 xx 10^(25)`
D. `4 xx 10^(25)`

Explanation

To find the number of atoms in 100 g of a face-centered cubic (FCC) crystal with a cell edge of 100 pm and a density of 10.0 g/cm³, we can follow these steps:

### Step 1: Convert the cell edge from picometers to centimeters
The cell edge (a) is given as 100 pm (picometers). We need to convert this to centimeters.

\[ 1 \text{ pm} = 10^{-12} \text{ m} \quad \text{and} \quad 1 \text{ m} = 100 \text{ cm} \]

Thus,

\[ 100 \text{ pm} = 100 \times 10^{-12} \text{ m} = 10^{-10} \text{ m} \]

Now converting to centimeters:

\[ 10^{-10} \text{ m} = 10^{-10} \times 100 \text{ cm} = 10^{-8} \text{ cm} \]

### Step 2: Calculate the volume of the unit cell
The volume (V) of the cubic unit cell can be calculated using the formula for the volume of a cube:

\[ V = a^3 \]

Substituting the value of a:

\[ V = (10^{-8} \text{ cm})^3 = 10^{-24} \text{ cm}^3 \]

### Step 3: Calculate the mass of one unit cell using density
Density (d) is given as 10.0 g/cm³. The mass (m) of one unit cell can be calculated using the formula:

\[ d = \frac{m}{V} \implies m = d \cdot V \]

Substituting the values:

\[ m = 10.0 \text{ g/cm}^3 \times 10^{-24} \text{ cm}^3 = 10^{-23} \text{ g} \]

### Step 4: Calculate the number of unit cells in 100 g
Now, we need to find out how many unit cells are present in 100 g of the crystal. We can use the mass of one unit cell calculated above:

\[ \text{Number of unit cells} = \frac{\text{Total mass}}{\text{Mass of one unit cell}} = \frac{100 \text{ g}}{10^{-23} \text{ g}} = 10^{25} \text{ unit cells} \]

### Step 5: Calculate the number of atoms in 100 g of the crystal
In an FCC unit cell, there are 4 atoms. Therefore, the total number of atoms in 100 g of the crystal can be calculated as:

\[ \text{Total number of atoms} = \text{Number of unit cells} \times \text{Number of atoms per unit cell} \]

Substituting the values:

\[ \text{Total number of atoms} = 10^{25} \text{ unit cells} \times 4 \text{ atoms/unit cell} = 4 \times 10^{25} \text{ atoms} \]

### Final Answer
The number of atoms in 100 g of this FCC crystal is \( 4 \times 10^{25} \) atoms.

---